Introduction to Statically
Indeterminate Analysis
Support reactions and internal
forces of statically determinate
structures can be determined
using only the equations of
equilibrium. However, the
analysis
y of statically
y indeterminate structures requires
additional equations based on
the geometry of deformation of
the structure.
1
Design of an indeterminate
structure is carried out in an
iterative manner, whereby the
(relative) sizes of the structural
members are initially assumed
and
d used
d to analyze
l
the
h structure.
Based on the computed results
(displacements and internal
member forces), the member
sizes are adjusted to meet
governing design criteria. This
iteration process continues until
the member sizes based on the
results of an analysis are close to
those assumed for that analysis.
3
Additional equations come from
compatibility relationships,
which ensure continuity of
displacements throughout the
structure. The remaining
equations are constructed from
member constitutive equations,
i.e., relationships between
stresses and strains and the
integration of these equations
o er the cross section
over
section.
2
Another consequence of
statically indeterminate
structures is that the relative
variation of member sizes
influences the magnitudes of
the forces that the member
will experience. Stated in
another way, stiffness (large
member size and/or high
modulus materials) attracts
force.
Despite these difficulties with
statically indeterminate
structures, an overwhelming
majority of structures being
built today are statically
4
indeterminate.
Also see pages 78 - 100 in your class notes.
1
Advantages Statically
Indeterminate Structures
5
Determinate beam is unstable
if middle support is removed
or knocked off!
7
Statically indeterminate
structures typically result in
smaller stresses and greater
stiffness (smaller deflections)
as illustrated for this beam.
6
Statically indeterminate
structures introduce redundancy,
which may insure that failure in
one part of the structure will not
result in catastrophic or collapse
failure of the structure.
8
2
Disadvantages of
Statically Indeterminate
Structures
Statically indeterminate structure
is self-strained due to support
settlement, which produces
stresses, as illustrated above.
9
10
Indeterminate Structures:
Influence Lines
Statically indeterminate structures are also self-strained due
to temperature changes and
fabrication errors.
11
Influence lines for statically
indeterminate structures
provide the same information
as influence lines for statically
determinate structures, i.e. it
represents the magnitude of a
response function at a
particular location on the
structure as a unit load moves
across the structure.
12
3
Our goals in this chapter are:
1.To become familiar with the
shape of influence lines for the
support reactions and internal
forces in continuous beams
and frames.
2.To develop an ability to sketch
the appropriate shape of
influence functions for
indeterminate beams and
frames.
a es
3.To establish how to position
distributed live loads on
continuous structures to
maximize response function
13
values.
The influence line for a force (or
moment) response function is
given by the deflected shape of
the released structure by
removing the displacement
constraint corresponding to the
response function of interest
from the original structure and
giving a unit displacement (or
rotation) at the location and in
the direction of the response
function.
15
Qualitative Influence
Lines for Statically Indeterminate Structures:
Muller-Breslau’s Principle
IIn many practical
i l applications,
li i
iit
is usually sufficient to draw only
the qualitative influence lines to
decide where to place the live
loads to maximize the response
functions of interest. The
Muller Breslau Principle pro
Muller-Breslau
provides a convenient mechanism
to construct the qualitative
influence lines, which is stated
as:
14
Procedure for constructing
qualitative influence lines for
indeterminate structures is: (1)
remove from the structure the
restraint corresponding
p
g to the
response function of interest, (2)
apply a unit displacement or
rotation to the released structure
at the release in the desired
response function direction, and
(3) draw the qualitative deflected
shape of the released structure
consistent with all remaining
support and continuity
conditions.
16
4
Notice that this procedure is
identical to the one discussed for
statically determinate structures.
However, unlike statically
d t
determinate
i t structures,
t t
the
th
influence lines for statically
indeterminate structures are
typically curved.
Placement of the live loads to
maximize the desired response
function is obtained from the
qualitative ILD.
17
QILD for RA
19
Uniformly distributed live
loads are placed over the
positive areas of the ILD to
maximize the drawn response
function values. Because the
influence line ordinates tend to
diminish rapidly with distance
from the response function
location, live loads placed more
than three span lengths away
can be ignored. Once the live
load pattern is known, an
indeterminate analysis of the
structure can be performed to
determine the maximum value of
the response function.
18
QILD’s for RC and VB
20
5
Live Load Pattern to
Maximize Forces in
Multistory Buildings
QILD’s for (MC)-,
(MD)+ and RF
21
maximum forces are
typically produced by
patterned loading.
Building codes specify that
members
b
off multistory
lti t
buildings be designed to
support a uniformly distributed
live load as well as the dead
load of the structure. Dead
and live loads are normally
considered separately since
the dead load is fixed in
position whereas the live load
must be varied to maximize a
particular force at each section
22
of the structure. Such
3. Axial column force (do not
consider axial force in beams):
Qualitative Influence Lines:
1. Introduce appropriate unit
displacement at the desired
response function location.
2. Sketch the displacement
diagram along the beam or
column line (axial force in
column) appropriate for the
unit displacement and
assume zero axial
deformation.
23
(a) Sketch the beam line
qualitative
lit ti di
displacement
l
t
diagrams.
(b) Sketch the column line
qualitative displacement
diagrams maintaining equality
of the connection geometry
before and after deformation.
24
6
4. Beam force:
(a) Sketch the beam line
qualitative displacement
diagram for which the release
has been introduced
introduced.
(b) Sketch all column line
qualitative displacement
diagrams maintaining
connection geometry before
and after deformation. Start
the column line qualitative
displacement diagrams from
the beam line diagram of (a).
(c) Sketch remaining beam
line qualitative displacement
diagrams maintaining connection geometry before and
after deformation.
25
27
26
Vertical
Reaction F
Load Pattern to
Maximize F
Column Moment
M
Load Pattern to
Maximize M 28
7
M
QILD and Load Pattern for
End Beam Moment M
QILD and Load Pattern for
Center Beam Moment M
Expanded Detail
for Beam End
Moment
29
Envelope Curves
Design engineers often use
influence lines to construct shear
and moment envelope
p curves for
continuous beams in buildings or
for bridge girders. An envelope
curve defines the extreme
boundary values of shear or
bending moment along the beam
due to critical placements of
design live loads. For example,
consider a three-span
continuous beam.
31
30
Qualitative influence lines for
positive moments are given,
shear influence lines are
presented later. Based on the
qualitative influence lines, critical
live load placement can be
determined and a structural
analysis computer program can
be used to calculate the member
end shear and moment values
for the dead load case and the
critical live load cases.
32
8
a
1
b
c
2
d
e
3
a
4
1
1
b
c
2
d
a
4
1
1
b
c
2
d
4
b
c
d
e
3
4
QILD for (Md)+
e
3
QILD for (Mb
e
3
2
QILD for (Ma)+
a
d
QILD for (Mc)+
e
3
c
2
Three-Span Continuous Beam
a
b
a
4
1
b
c
2
)+
d
e
3
4
QILD for (Me)+
33
a
1
b
c
2
d
34
a
e
3
4
1
Critical Live Load Placement
for (Ma)+
a
1
b
2
c
d
c
2
d
e
3
4
Critical Live Load Placement
for (Mb)+
e
3
b
a
4
Critical Live Load Placement
for (Ma)-
1
b
2
c
d
e
3
4
Critical Live Load Placement
for (Mb)35
36
9
a
1
b
c
2
d
e
3
a
4
1
Critical Live Load Placement
for (Mc)+
a
1
b
c
2
d
c
2
d
e
3
4
Critical Live Load Placement
for (Md)+
e
3
b
a
4
1
Critical Live Load Placement
for (Mc)-
b
c
2
d
e
3
4
Critical Live Load Placement
for (Md)37
a
1
b
c
2
d
Calculate the moment envelope
curve for the three-span
continuous beam.
e
3
38
4
a
Critical Live Load Placement
for (Me)+
1
1
b
2
c
d
e
3
4
Critical Live Load Placement
for (Me)39
c
2
L
a
b
d
e
3
L
4
L
L = 20’ = 240”
E = 3,000 ksi
A = 60 in2
I = 500 in4
wDL = 1.2 k/ft – dead load
wLL = 4.8 k/ft – live load
40
10
Shear and Moment Equations
for a Loaded Span
Mi
q
Load Cases
wDL
Mie
a
xi
1
Vie
Vi
Shear and Moment Equations
q
for an Unloaded Span
1
(set q = 0 in equations above)
a
Vie = Vi
1
Mie = -Mi + Vi xi
1
b
c
d
e
2
LC4
3
wLL
b
c
d
e
2
LC5
3
wLL
wLL
a
1
4
1
c
d
2
LC6
3
2
3
b
c
d
2
LC2
wLL
3
b
c
d
2
LC3
3
c
LC7
4
e
4
e
4
A summary of the results from
the statically indeterminate beam
analysis for each of the seven
load cases are given in your
class notes.
----- RESULTS FOR LOAD SET: 1
***** M E M B E R F O R C E S *****
MEMBER
MEMBER NODE
b
b
LC1
42
4
e
d
3
AXIAL
FORCE
(kip)
SHEAR
FORCE
(kip)
BENDING
MOMENT
(ft-k)
1
1
2
0.00
-0.00
9.60
14.40
0.00
-48.00
2
2
3
0.00
-0.00
12.00
12.00
48.00
-48.00
3
3
4
0.00
-0.00
14.40
9.60
48.00
0.00
4
wLL
a
2
e
41
wLL
a
d
wLL
a
Mie = -Mi + Vi xi – 0.5q (xi)2
1
c
wLL
Vie = Vi – q xi
a
b
e
443
44
11
Load Case 2
The equations for the internal
shear and bending moments for
each span and each load case
are:
Load Case 1
V12 = 43.2 – 4.8x1
M12 = 43.2x1 – 2.4(x1)2
V23 = 0
M23 = -96
96
V34 = 52.8 – 4.8x3
M34 = -96 + 52.8x3 – 2.4(x3)2
V12 = 9.6 – 1.2x1
M12 = 9.6x1 – 0.6(x1)2
Load Case 3
V23 = 12 – 1.2x2
M23 = -48
48 + 12x2 – 0.6(x
0 6(x2)2
V12 = -4.8
M12 = -4.8x
48 1
V34 = 14.4 – 1.2x3
M34 = -48 + 14.4x3 – 0.6(x3)2
V23 = 48 – 4.8x2
M23 = -96 + 48x2 – 2.4(x2)2
45
Load Case 4
V34 = 4.8
M34 = -96 + 4.8x3
46
Load Case 6
V12 = 41.6 – 4.8x1
M12 = 41.6x1 – 2.4(x1)2
V12 = 36.8 – 4.8x1
M12 = 36.8x1 – 2.4(x1)2
V23 = 8
M23 = -128
128 + 8x2
V23 = 56 – 4.8x2
M23 = -224
224 + 56x2 – 2.4(x
2 4(x2)2
V34 = -1.60
M34 = 32 - 1.6x3
V34 = 3.2
M34 = -64 + 3.2x3
Load Case 5
Load Case 7
V12 = 1.6
M12 = 1.6x
16 1
V12 = -3.2
M12 = -3.2x
32 1
V23 = -8
M23 = 32 - 8x2
V23 = 40 – 4.8x2
M23 = -64 + 40x2 – 2.4(x2)2
V34 = 54.4 – 4.8x3
47
M34 = -128 + 54.4x3 – 2.4(x3)2
V34 = 59.2 – 4.8x3
48
M34 = -224 + 59.2x3 – 2.4(x3)2
12
Bending Moment Diagram LC1
49
50
51
52
13
A spreadsheet program listing is
included in your class notes that
gives the moment values along
the span lengths and is used to
graph the moment envelope
curves.
curves
In the spreadsheet:
Live Load E-Mom (+)
= max (LC2 through LC7)
Live Load E-Mom (-)
= min (LC2 through LC7)
Live Load E-Mom (+)
Live Load E-Mom (-)
53
Total Load E-Mom (+) = LC1
+ Live Load E-Mom (+)
Total Load E-Mom (-) = LC1
+ Live Load E-Mom (-)
54
Construction of the shear
envelope curve follows the same
procedure. However,, jjust as is
p
the case with a bending moment
envelope, a complete analysis
should also load increasing/
decreasing fractions of the span
where shear is being considered.
Total Load E-Mom (+)
Total Load E-Mom (-)
55
56
14
1
a
a
b
c
d
1
1
2
b
c
3
2
b
1
c
2
d
1
a
4
QILD (V2
4
-1 QILD (V L)+
3
e
3
-1
e
3
4
QILD (V1)+
a
d
e
1
b
c
2
d
e
3
4
QILD (V3R)+
L)+
1
a
a
b
c
d
1
1
2
3
b
c
d
e
e
2
4
QILD (V2R)+
57
Shear ILD Notation:
Superscript L = just to the left of
the subscript point
Superscript R = just to the right
of the subscript point
To obtain the negative shear
qualitative influence line diagrams simply flip the drawn
positive qualitative influence line
diagrams.
59
QILD (V4)+
3
4
-1
58
In practice, the construction of the
exact shear envelope is usually
unnecessary since an approximate
envelope obtained by connecting
the maximum possible shear at the
reactions with the maximum
possible value at the center of the
spans is sufficiently accurate. Of
course, the dead load shear must
be added to the live load shear
envelope.
p
60
15